Introduction

In the last chapter we established the most natural version of the qualitative theory of logarithmic forms in the context of algebraic groups. Many of the most important applications, however, involve a quantitative form of the theory and this we shall discuss in the present chapter. We shall begin with a report on the results concerning linear forms in ordinary logarithms which refine the basic theory as described in Chapter 2. The estimates given here are fully explicit and they are considerably sharper than those described previously; their derivation depends critically on the theory of multiplicity estimates on group varieties in the form given in Chapter 5. In the following section we report on generalisations to logarithms related to arbitrary commutative algebraic groups. The best general results to date are due to Hirata-Kohno, and more recently Gaudron, and the precision of these is now quite close to those obtainable in the classical case. The work here arises from a long series of earlier researches beginning with publications of Baker and Masser in the elliptic and abelian cases and subsequently taken up especially by Coates, Lang, Philippon and Waldschmidt. This has been a very active area of study and there are, in particular, some significant further contributions in the elliptic case by S. David.